Properties

Label 338.4.a.c
Level $338$
Weight $4$
Character orbit 338.a
Self dual yes
Analytic conductor $19.943$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{3} + 4 q^{4} + 18 q^{5} - 8 q^{6} - 20 q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{3} + 4 q^{4} + 18 q^{5} - 8 q^{6} - 20 q^{7} - 8 q^{8} - 11 q^{9} - 36 q^{10} + 48 q^{11} + 16 q^{12} + 40 q^{14} + 72 q^{15} + 16 q^{16} + 66 q^{17} + 22 q^{18} + 16 q^{19} + 72 q^{20} - 80 q^{21} - 96 q^{22} + 168 q^{23} - 32 q^{24} + 199 q^{25} - 152 q^{27} - 80 q^{28} + 6 q^{29} - 144 q^{30} - 20 q^{31} - 32 q^{32} + 192 q^{33} - 132 q^{34} - 360 q^{35} - 44 q^{36} - 254 q^{37} - 32 q^{38} - 144 q^{40} + 390 q^{41} + 160 q^{42} - 124 q^{43} + 192 q^{44} - 198 q^{45} - 336 q^{46} + 468 q^{47} + 64 q^{48} + 57 q^{49} - 398 q^{50} + 264 q^{51} + 558 q^{53} + 304 q^{54} + 864 q^{55} + 160 q^{56} + 64 q^{57} - 12 q^{58} + 96 q^{59} + 288 q^{60} - 826 q^{61} + 40 q^{62} + 220 q^{63} + 64 q^{64} - 384 q^{66} + 160 q^{67} + 264 q^{68} + 672 q^{69} + 720 q^{70} + 420 q^{71} + 88 q^{72} - 362 q^{73} + 508 q^{74} + 796 q^{75} + 64 q^{76} - 960 q^{77} + 776 q^{79} + 288 q^{80} - 311 q^{81} - 780 q^{82} - 320 q^{84} + 1188 q^{85} + 248 q^{86} + 24 q^{87} - 384 q^{88} - 1626 q^{89} + 396 q^{90} + 672 q^{92} - 80 q^{93} - 936 q^{94} + 288 q^{95} - 128 q^{96} + 1294 q^{97} - 114 q^{98} - 528 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 4.00000 4.00000 18.0000 −8.00000 −20.0000 −8.00000 −11.0000 −36.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.a.c 1
13.b even 2 1 26.4.a.c 1
13.c even 3 2 338.4.c.e 2
13.d odd 4 2 338.4.b.d 2
13.e even 6 2 338.4.c.a 2
13.f odd 12 4 338.4.e.a 4
39.d odd 2 1 234.4.a.e 1
52.b odd 2 1 208.4.a.b 1
65.d even 2 1 650.4.a.b 1
65.h odd 4 2 650.4.b.f 2
91.b odd 2 1 1274.4.a.d 1
104.e even 2 1 832.4.a.d 1
104.h odd 2 1 832.4.a.o 1
156.h even 2 1 1872.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 13.b even 2 1
208.4.a.b 1 52.b odd 2 1
234.4.a.e 1 39.d odd 2 1
338.4.a.c 1 1.a even 1 1 trivial
338.4.b.d 2 13.d odd 4 2
338.4.c.a 2 13.e even 6 2
338.4.c.e 2 13.c even 3 2
338.4.e.a 4 13.f odd 12 4
650.4.a.b 1 65.d even 2 1
650.4.b.f 2 65.h odd 4 2
832.4.a.d 1 104.e even 2 1
832.4.a.o 1 104.h odd 2 1
1274.4.a.d 1 91.b odd 2 1
1872.4.a.q 1 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} - 4 \) Copy content Toggle raw display
\( T_{5} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 4 \) Copy content Toggle raw display
$5$ \( T - 18 \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T - 48 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 66 \) Copy content Toggle raw display
$19$ \( T - 16 \) Copy content Toggle raw display
$23$ \( T - 168 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 20 \) Copy content Toggle raw display
$37$ \( T + 254 \) Copy content Toggle raw display
$41$ \( T - 390 \) Copy content Toggle raw display
$43$ \( T + 124 \) Copy content Toggle raw display
$47$ \( T - 468 \) Copy content Toggle raw display
$53$ \( T - 558 \) Copy content Toggle raw display
$59$ \( T - 96 \) Copy content Toggle raw display
$61$ \( T + 826 \) Copy content Toggle raw display
$67$ \( T - 160 \) Copy content Toggle raw display
$71$ \( T - 420 \) Copy content Toggle raw display
$73$ \( T + 362 \) Copy content Toggle raw display
$79$ \( T - 776 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 1626 \) Copy content Toggle raw display
$97$ \( T - 1294 \) Copy content Toggle raw display
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